for all \(y,z\in K(G,N)\) , where \(\Delta \)  is the modular function. We refer to [1]} page 278 for more details about \(W^*\) -crossed products with locally compact groups. Consider a faithful normal semifinite weight \(\varphi \)  on \(N\) . Now using theorem \(3.1\)  of [2]}, we have the dual weight \(\tilde{\varphi }\)  which is such that for \(x\in K(G,N)\)  we have :
\(\tilde{\varphi }((\pi _{\alpha } \times \lambda )(x^*\times x))=\varphi ((x^*\times x)(e)).\) 
